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Chapter 2
Optical Transitions
2.1 Introduction
Among energy states, the state with the lowest energy is most stable. Therefore, the
electrons in semiconductors tend to stay in low energy states. If they are excited
by thermal energy, light, or electron beams, the electrons absorb these energies and
transit to high energy states. These transitions of the electrons from low energy states
to high energy states are called excitations. High energy states, however, are unstable.
As a result, to take stable states, the electrons in high energy states transit to low
energy states in certain lifetimes. These transitions of the excited electrons from high
energy states to low energy states are referred to as relaxations. The excitation and
relaxation processes between the valence band and the conduction band are shown
in Fig. 2.1.
In semiconductors, the transitions of electrons from high energy states to low
energy states are designated recombinations of the electrons and the holes. In the
recombinations of the electrons and the holes, there are radiative recombinations
and nonradiative recombinations. The radiative recombinations emit photons, and
the energies of the photons correspond to a difference in the energies between the
initial and final energy states related to the transitions. In contrast, in the nonradiative recombinations, the phonons are emitted to crystal lattices or the electrons are
trapped in the defects, and the transition energy is transformed into forms other than
light. The Auger processes are also categorized as nonradiative recombinations. To
obtain high efficiency semiconductor light emitting devices, we have to minimize
the nonradiative recombinations. However, to enhance modulation characteristics,
the nonradiative recombination centers may be intentionally induced in the active
layers, because they reduce the carrier lifetimes (see Sect. 5.1).
Let us consider the transitions of the electrons from the bottom of the conduction
band to the top of the valence band. A semiconductor, in which the bottom of the
conduction band and the top of the valence band are placed at a common wave vector
k, is the direct transition semiconductor. A semiconductor, in which the bottom
of the conduction band and the top of the valence band have different k-values,
© Springer Japan 2015
T. Numai, Fundamentals of Semiconductor Lasers,
Springer Series in Optical Sciences 93, DOI 10.1007/978-4-431-55148-5_2
23
24
Fig. 2.1 Excitation and
relaxation
2 Optical Transitions
Conduction
band
Ec
Electron
Ev
Electron
Hole
Hole
Electron
Valence
band
Stable state
Fig. 2.2 a Direct and b
indirect transition semiconductors
Electron
(a)
Excitation
Relaxation
(b)
E
Stable state
E
Ec
Ec
Ev
0
k
Ev
0
k
is the indirect transition semiconductor. These direct and indirect transitions are
schematically shown in Fig. 2.2. In transitions of the electrons, the energy and the
momentum are conserved, respectively. Therefore, the phonons do not take part in
direct transitions. Because the wave vector k of the phonons is much larger than
that of the photons, the phonon transitions accompany the indirect transitions to
satisfy the momentum conservation law. Hence, in the direct transitions, the transition
probabilities are determined by only the electron transition probabilities. In contrast,
in the indirect transitions, the transition probabilities are given by a product of the
electron transition probabilities and the phonon transition probabilities. As a result,
the transition probabilities of the direct transitions are much higher than those of the
indirect transitions. Consequently, the direct transition semiconductors are superior
to the indirect ones for light emitting devices.
2.2 Light Emitting Processes
Light emission due to the radiative recombinations is called the luminescence.
According to the lifetime, the excitation methods, and the energy states related to
the transitions, light emitting processes are classified as follows.
2.2 Light Emitting Processes
25
2.2.1 Lifetime
With regard to the lifetime, there are two light emissions: fluorescence, with a short
lifetime of 10−9 –10−3 s, and phosphorescence, with a long lifetime of 10−3 s to one
day.
2.2.2 Excitation
Luminescence due to optical excitation (pumping) is photoluminescence, which is
widely used to characterize materials. Optical excitation is also used to pump dye
lasers (for example, Rhodamine 6G and Coumalin) and solid-state lasers (e.g., YAG
and ruby). When the photon energy of the pumping light is ω1 and that of the
luminescence is ω2 , the luminescence with ω2 < ω1 is called Stokes luminescence and that with ω2 > ω1 is designated anti-Stokes luminescence. Luminescence caused by electrical excitation is electroluminescence, which has been used for
panel displays. In particular, luminescence by current injection is called injectiontype electroluminescence; it has been used for light emitting diodes (LEDs) and
semiconductor lasers or laser diodes (LDs). In such injection-type optical devices,
the carriers are injected into the active layers by forward bias across the pn junctions.
Note that the current (carrier) injection is also considered the excitation, because it
generates a lot of high energy electrons. The luminescence due to electron beam irradiation is cathodoluminescence, which has been adopted to characterize materials.
The luminescence induced by mechanical excitation using stress is triboluminescence, and that by thermal excitation is thermoluminescence. Luminescence during
a chemical reaction is referred to chemiluminescence; it has not been reported in
semiconductors.
2.2.3 Transition States
Figure 2.3 shows light emission processes between various energy states. They
are classified into impurity recombinations, interband recombinations, and exciton
recombinations.
In impurity recombinations, there is recombination between the electron in the
conduction band and the empty acceptor level with the photon energy of ωA , recombination between the electron in the donor level and the hole in the valence band with
the photon energy of ωD , and recombination between the electron in the donor level
and the empty acceptor level with the photon energy of ωDA . By observing light
emissions due to these recombinations at extremely low temperatures, information
on doping characteristics is obtained.
26
2 Optical Transitions
Fig. 2.3 Light emission
processes
In interband recombinations between the conduction and valence bands, light
emission in the vicinity of the band edges with the photon energy of ωg is dominant.
This band edge emission is used in most LEDs and LDs composed of III–V group
semiconductors.
The exciton recombination, with the photon energy of ωe , is the recombination
of the electron generated by decay of an exciton and the hole in the valence band.
2.3 Spontaneous Emission, Stimulated Emission, and Absorption
Figure 2.4 schematically shows radiations and absorption. In the radiations, there
are spontaneous emission and stimulated emission (or induced emission). As shown
in Fig. 2.4a, spontaneous emission is a radiative process in which an excited electron
decays in a certain lifetime and a photon is emitted. Note that spontaneous emission
takes place irrespective of incident lights. In contrast, in the stimulated emission
illustrated in Fig. 2.4b, an incident light induces a radiative transition of an excited
electron. The emitted light due to the stimulated emission has the same wavelength,
phase, and direction as the incident light. Therefore, the light generated by the stimulated emission is highly monochromatic, coherent, and directional. In the stimulated
emission, one incident photon generates two photons; one is the incident photon
itself, and the other is an emitted photon due to the stimulated emission. As a result,
the incident light is amplified by the stimulated emission. The absorption depicted in
Fig. 2.4c is a process that the electron transits from a lower energy state to a higher
one by absorbing energy from the incident light. Because this transition is induced
by the incident light, it is sometimes called induced absorption. It should be noted
that spontaneous absorption does not exist; this will be explained in Chap. 8.
When a light is incident on a material, the stimulated emission and the absorption
simultaneously take place. In thermal equilibrium, there are more electrons in a lower
energy state than in a higher one, because the lower energy state is more stable than
the higher one. Therefore, in thermal equilibrium, only the absorption is observed
when a light is incident on a material. In order to obtain a net optical gain, we have
to make the number of electrons in a higher energy state larger than in the lower one.
This condition is referred to as inverted population, or, population inversion because
the electron population is inverted compared with that in thermal equilibrium. In
2.3 Spontaneous Emission, Stimulated Emission, and Absorption
Fig. 2.4 Radiation and
absorption: a spontaneous
emission, b stimulated emission, and c absorption
(a)
(b)
27
(c)
Ec
Ev
semiconductors, the population inversion is obtained only in the vicinity of the band
edges by excitation of the electrons through optical pumping or electric current
injection. The population inversion generates many electrons at the bottom of the
conduction band and many holes at the top of the valence bands.
The laser oscillators use fractions of the spontaneous emission as the optical
input and amplify the fractions by the stimulated emission under population inversion. Once the optical gains exceed the optical losses in the laser oscillators, laser
oscillations take place. The term laser is an acronym for “light amplification by
stimulated emission of radiation” and is used as a noun in a laser oscillator. Note
that an emitted light from a laser oscillator is not a laser but a laser light or a laser
beam. As a back formation from laser, “lase” is used as a verb meaning “to emit
coherent light.”
Among semiconductor light emitting devices, LEDs use spontaneous emission
and are applied to remote-control transmitters, switch lights, brake lights, displays,
and traffic signals. In contrast, LDs are oscillators of lights using stimulated emission
and are used as light sources for lightwave communications, compact discs (CDs),
magneto-optical discs (MOs), digital video discs (DVDs), laser beam printers, laser
pointers, bar-code readers, and so on.
2.4 Optical Gains
2.4.1 Lasers
Let us consider the relationships between laser oscillators and optical gains. First,
we review a general oscillator, as shown in Fig. 2.5. An oscillator has a gain and
amplifies an input. Also, the oscillator returns a fraction of an output to an input port
through a feedback loop. The returned one is repetitively amplified, and oscillation
starts when a net gain exceeds an internal loss of the oscillator.
Figure 2.6 shows a laser oscillator where a fraction of the spontaneous emission is
used as the input and the optical gain is produced by the stimulated emission. To feed
back light, optical resonators or optical cavities, which are composed of reflectors or
mirrors, are adopted. Due to this configuration, characteristics of lasers (for brevity,
28
2 Optical Transitions
Fig. 2.5 Oscillator
β
Output
Input
Gain
+
Fig. 2.6 Laser oscillator
β
Fraction of
spontaneous
emission
+
Feedback
Optical
gain
Optical feedback by
reflectors
Laser light
lasers are used for laser oscillators here) are affected by the optical gains and optical
resonators. Many lasers use optical resonators, but nitrogen lasers, whose optical
gains are very large, can start laser oscillations even without optical resonators. It
should be noted that all semiconductor lasers use optical resonators.
As explained earlier, a fraction of the spontaneous emission is used as the input
of a laser. Note that all the spontaneously emitted lights cannot be used as the input,
because they have different wavelengths, phases, and propagation directions. Among
these lights, only the lights, which have wavelengths within the optical gain spectrum
and satisfy the resonance conditions of the optical resonators, can be the sources of
laser oscillations. Other spontaneous emissions, which do not satisfy the resonance
conditions of optical resonators, are readily emitted outward without obtaining sufficient optical gains for laser oscillations. The lights amplified by the stimulated
emission have the same wavelength, phase, and propagation direction as those of the
input lights. Therefore, the laser lights are highly monochromatic, bright, coherent,
and directional.
2.4.2 Optical Gains
2.4.2.1 Carrier Distribution
We derive equations for the optical gains of semiconductor lasers. We suppose that
many electrons are excited to the conduction band by optical pumping or electrical
current injection. In this condition, the carrier distribution is in nonthermal equilibrium, and there are many electrons in the conduction band and many holes in the
valence band. As a result, one Fermi level E F cannot describe the distribution functions of the electrons and holes. In this case, it is useful to determine the distribution
functions by assuming that the electrons in the conduction band and the holes in the
2.4 Optical Gains
29
valence band are separately governed by Fermi–Dirac distribution. For this purpose,
we introduce quasi-Fermi levels E Fc and E Fv defined as
E c − E Fc
n = Nc exp −
,
kB T
E Fv − E v
p = Nv exp −
,
kB T
2 π m e ∗ kB T 3/2
,
Nc = 2
h2
2 π m h ∗ kB T 3/2
Nv = 2
.
h2
(2.1)
Here, n and p are the electron concentration and the hole concentration, respectively;
E c and E v are the energy of the bottom of the conduction band and that of the top
of the valence band, respectively; kB = 1.3807 × 10−23 J/K is Boltzmann constant;
T is an absolute temperature; Nc and Nv are the effective density of states for the
electrons and that for the holes, respectively; m e ∗ and m h ∗ are the effective mass of
the electrons and that of the holes, respectively; and h is Planck’s constant.
From (2.1), these quasi-Fermi levels E Fc and E Fv are written as
n
,
Nc
p
.
= E v − kB T ln
Nv
E Fc = E c + kB T ln
E Fv
(2.2)
We express a distribution function for the electrons in the valence band with the
energy E 1 as f 1 and that for the electrons in the conduction band with the energy E 2
as f 2 . Using E Fc and E Fv , we can express f 1 and f 2 as
f1 =
1
,
exp [(E 1 − E Fv ) / (kB T )] + 1
(2.3)
1
f2 =
.
exp [(E 2 − E Fc ) / (kB T )] + 1
It should be noted that the distribution function for the holes in the valence band is
given by [1 − f 1 ].
2.4.2.2 Optical Transition Rates
As shown in Fig. 2.7, we assume that a light, which has a photon energy of
E 21 =E 2 −E 1 and a photon density of n ph (E 21 ), interacts with a direct-transition
30
2 Optical Transitions
Fig. 2.7 Schematic model for
radiation and absorption
E2
Ec
nph (E21)
Ev
E1
semiconductor. Under this assumption, we calculate the transition rates for the stimulated emission, the absorption, and the spontaneous emission. In Fig. 2.7, E c is the
energy for the bottom of the conduction band, and E v is the energy for the top of the
valence band. Note that this figure shows the energy bands for a certain value of k,
and the horizontal line has no physical meaning.
(i) Stimulated Emission Rate
The stimulated emission rate per unit volume r21 (stim) is given by a product of
the transition probability per unit time from E 2 to E 1 : B21 ,
the electron concentration in a state with the energy E 2 : n 2 ,
the hole concentration in a state with the energy E 1 : p1 ,
the photon density: n ph (E 21 ).
The concentration n 2 of the electron, which occupies a state with the energy E 2
in the conduction band, is expressed as
n 2 = ρc (E 2 − E c ) f 2 ,
(2.4)
where ρc (E 2 − E c ) is the density of states, which is a function of E 2 − E c , and
f 2 is the distribution function in (2.3).
The concentration p1 of the hole, which occupies a state with the energy E 1 in
the valence band, is written as
p1 = ρv (E v − E 1 )[1 − f 1 ],
(2.5)
where ρv (E v − E 1 ) is the density of states, which is a function of E v − E 1 , and
f 1 is the distribution function in (2.3).
Therefore, the stimulated emission rate r21 (stim) is obtained as
r21 (stim) = B21 n 2 p1 n ph (E 21 )
= B21 n ph (E 21 )ρc (E 2 − E c )ρv (E v − E 1 ) f 2 [1 − f 1 ].
(2.6)
2.4 Optical Gains
31
(ii) Absorption Rate
The absorption rate per unit volume r12 (abs) is given by a product of
the transition probability per unit time from E 1 to E 2 : B12 ,
the concentration of an empty state with the energy E 2 : p2 ,
the electron concentration in a state with the energy E 1 : n 1 ,
the photon density: n ph (E 21 ).
The concentration p2 of an empty state, which is not occupied by the electrons
with the energy E 2 in the conduction band, is expressed as
p2 = ρc (E 2 − E c )[1 − f 2 ].
(2.7)
The concentration n 1 of the electron, which occupies a state with the energy E 1
in the valence band, is written as
n 1 = ρv (E v − E 1 ) f 1 .
(2.8)
As a result, the absorption rate r12 (abs) is obtained as
r12 (abs) = B12 p2 n 1 n ph (E 21 )
= B12 n ph (E 21 )ρc (E 2 − E c )ρv (E v − E 1 ) f 1 [1 − f 2 ].
(2.9)
(iii) Spontaneous Emission Rate
The spontaneous emission rate per unit volume r21 (spon) is independent of the
incident photon density and given by a product of
the transition probability per unit time from E 2 to E 1 : A21 ,
the electron concentration in a state with the energy E 2 : n 2 ,
the hole concentration in a state with the energy E 1 : p1 .
Using (2.4) and (2.5), the spontaneous emission rate r21 (spon) is obtained as
r21 (spon) = A21 n 2 p1
= A21 ρc (E 2 − E c )ρv (E v − E 1 ) f 2 [1 − f 1 ].
(2.10)
2.4.2.3 Optical Transition in Thermal Equilibrium
In thermal equilibrium, the carrier distributions are described by one Fermi level E F ,
and the radiations balance the absorption. In other words, a sum of the stimulated
32
2 Optical Transitions
emission rate and the spontaneous emission rate is equal to the absorption rate. Using
the optical transition rates for the stimulated emission, the spontaneous emission, and
the absorption, we obtain
r21 (stim) + r21 (spon) = r12 (abs),
E Fc = E Fv = E F .
(2.11)
Substituting (2.6), (2.9), and (2.10) into (2.11), we have
n ph (E 21 ) =
A21
,
B12 exp[E 21 /(kB T )] − B21
(2.12)
while the blackbody radiation theory gives
n ph (E 21 ) =
8 π n r 3 E 21 2
.
h 3 c3 exp[E 21 /(kB T )] − h 3 c3
(2.13)
Here, n r is the effective refractive index of a material, h is Planck’s constant, and
c = 2.99792458 × 108 m/s is the speed of light in a vacuum.
Comparison of (2.12) and (2.13) results in
A21 = Z (E 21 )B, B21 = B12 = B,
8 π n r 3 E 21 2
Z (E 21 ) =
,
h 3 c3
(2.14)
which is known as Einstein’s relation, and A21 is called Einstein’s A coefficient; and
B is called Einstein’s B coefficient. It should be noted that B21 = B12 = B, and A21
is proportional to B.
We consider the physical meaning of Z (E 21 ) in (2.14) in the following. We suppose that a light field E L is written as
E L = A0 exp[ i (ωt − k · r)],
(2.15)
where A0 is a complex amplitude, ω is an angular frequency, t is a time, k is a
wave vector, and r is a position vector. We also suppose that the periodic boundary
condition is satisfied on the surfaces of a cube with a side length L. Because the
electric fields at x = 0 and x = L are the same, an eigenmode is given by
exp(0) = exp( i k x L) = 1, ∴ k x =
2π
nx ,
L
(2.16)
where k x is an x-component of the wave vector k and n x is an integer. Similarly, we
obtain
2π
2π
n y , kz =
nz ,
(2.17)
ky =
L
L
2.4 Optical Gains
33
where n y and n z are integers. Hence, k 2 = k x 2 + k y 2 + k z 2 is expressed as
k2 =
2π
L
2
(n x 2 + n y 2 + n z 2 ).
(2.18)
Using the effective refractive index n r of the material, the angular frequency ω is
given by
kc
(2.19)
ω= .
nr
Therefore, the energy E = ω is written as
E = ω =
2 π c
hc
(n x 2 + n y 2 + n z 2 )1/2 =
(n x 2 + n y 2 + n z 2 )1/2 ,
nr L
nr L
(2.20)
which takes discrete values, because n x , n y , and n z are integers. However, when L
is much larger than the wavelength of the light, n x 2 + n y 2 + n z 2 becomes a huge
number, and many modes exist near each other. When we consider a sphere with a
radius of R = (n x 2 + n y 2 + n z 2 )1/2 , the number of combinations (n x , n y , n z ) is
equal to a volume of the sphere 4 π R 3 /3. For each combination (n x , n y , n z ), there
exist two modes corresponding to their polarizations, which are perpendicular to
each other. As a result, the number of modes whose energy is placed between 0 and
E is given by
8 π nr 3 E 3 3
4 π nr E L 3
=
L .
(2.21)
2×
3
hc
3h 3 c3
The number of modes with their energies between E and E + dE is obtained as
a derivative of (2.21) with respect to E, which is expressed as
8 π nr 3 E 2 3
L dE.
h 3 c3
(2.22)
Because a volume of the cube is L 3 , dividing (2.22) by L 3 gives the number of modes
per unit volume, that is, the mode density m(E) dE as
m(E) dE =
8 π nr 3 E 2
dE.
h 3 c3
(2.23)
From (2.23) and the third equation in (2.14), it is found that we have m(E 21 ) =
Z (E 21 ). This relation suggests that the spontaneous emission takes place for all the
modes with their energy placed between E and E +dE while the stimulated emission
and absorption occur only for the mode corresponding to the incident light.
Note that the derivation of A and B assumes that both r21 (stim) and r12 (abs) are
proportional to the photon density n ph (E 21 ). If we assume that both r21 (stim) and
r12 (abs) are proportional to the energy density of the photons as Einstein did [1],
other formulas for A and B will be obtained.
34
2 Optical Transitions
2.4.2.4 Definitions of the Mode Density
In the preceding discussion, we explained the mode density m(E) dE with the energies placed between E and E + dE. Here, we will derive mode densities m(ω) dω
with the angular frequencies of a light between ω and ω + dω, and m(ν) dν with the
light frequencies between ν and ν + dν. Using the (effective) refractive index of the
material n r and the speed of light in a vacuum c, we have
k=
nr ω
2 π nr ν
=
.
c
c
(2.24)
Substituting (2.24) into (2.18) and then calculating a volume 4 π R 3 /3 of a sphere
with the radius R = (n x 2 + n y 2 + n z 2 )1/2 results in
2×
4π 3
nr 3 ω3 3
8 π nr 3ν 3 3
L =
L ,
R =
2
3
3
3π c
3c3
(2.25)
where a factor 2 corresponds to the directions of the polarizations. As a result, dividing
(2.25) by L 3 and then differentiating with respect to ω or ν leads to
m(ω) dω =
nr 3 ω2
8 π nr 3ν 2
dω,
m(ν)
dν
=
dν.
π2 c 3
c3
(2.26)
2.4.2.5 Net Stimulated Emission
Let us consider the excited conditions where many free carriers exist. In these nonthermal equilibrium conditions, radiations do not balance with absorption anymore.
When a light is incident on a material, the stimulated emission and the absorption
simultaneously take place. Therefore, the net stimulated emission rate r 0 (stim) is
given by
r 0 (stim) = r21 (stim) − r12 (abs)
= Bn ph (E 21 )ρc (E 2 − E c )ρv (E v − E 1 )[ f 2 − f 1 ]
A21
n ph (E 21 )ρc (E 2 − E c )ρv (E v − E 1 )[ f 2 − f 1 ].
=
Z (E 21 )
(2.27)
From (2.27), to obtain the net stimulated emission r 0 (stim) > 0, we need
f2 > f1 ,
(2.28)
which indicates the population inversion in the semiconductors. With the help of
(2.3), (2.28) reduces to
2.4 Optical Gains
35
E Fc − E Fv > E 2 − E 1 = E 21 ,
(2.29)
which is known as the Bernard–Duraffourg relation. In semiconductor lasers, a
typical carrier concentration for r 0 (stim) > 0 is of the order of 1018 cm−3 .
2.4.2.6 Net Stimulated Emission Rate and Optical Gains
The optical power gain coefficient per unit length g is defined as
dI
= g I,
dz
(2.30)
where I is the light intensity per unit area and z is a coordinate for a propagation
direction of the light. Because the light intensity is proportional to the square of the
electric field, the amplitude gain coefficient gE , which is the gain coefficient for the
electric field, is given by gE = g/2. From the definition, I is expressed as
I = v E 21 n ph (E 21 ) = v ωn ph (E 21 ),
dω
c
v=
= ,
dk
nr
dn ph (E 21 )
dI
= v E 21
= v E 21 r 0 (stim),
dt
dt
(2.31)
where v is a group velocity of the light in the material, E 21 = ω is the photon
energy, and n ph (E 21 ) is the photon density.
Here we consider a relationship between the derivatives of I with respect to a
position and a time. Because the position z is a function of the time t, we have
dt dI
dI
=
=
dz
dz dt
dz
dt
−1
1 dI
dI
=
,
dt
v dt
(2.32)
dI
= g I = g v E 21 n ph (E 21 ).
dz
From (2.31) and (2.32), we have the relation
r 0 (stim) = v g n ph (E 21 ).
(2.33)
As a result, it is found that a large r 0 (stim) leads to a large g. Using (2.27), (2.31),
and (2.33), we can also express g as
g=
nr
r 0 (stim) n r
= Bρc (E 2 − E c )ρv (E v − E 1 )[ f 2 − f 1 ].
n ph (E 21 ) c
c
(2.34)
36
2 Optical Transitions
From the time-dependent quantum mechanical perturbation theory (see Appendix C), Einstein’s B coefficient in (2.14), which is the transition rate due to the
stimulated emission for a photon in a free space, is given by
B=
e2 h
|1|p|2|2 .
2m 2 ε0 n r 2 E 21
(2.35)
Here, e is the elementary charge; h is Planck’s constant; m is the electron mass; ε0 is
permittivity in a vacuum; n r is the refractive index of the material; E 21 is the photon
energy; p is the momentum operator; and |1 and |2 are the wave functions of the
valence band and the conduction band in a steady state, respectively.
Substituting (2.35) into (2.14), we can write Einstein’s A coefficient as
A21 =
4 π e2 n r E 21
|1|p|2|2 .
m 2 ε0 h 2 c 3
(2.36)
Using the dipole moment μ, (2.35) and (2.36) can be rewritten as follows: The
momentum operator p is expressed as
p=m
dr
.
dt
(2.37)
As a result, we obtain
1
d
1|p|2 =
1|r|2 = i ω1|r|2,
m
dt
(2.38)
where r ∝ e i ωt was assumed. With the help of (2.38), Einstein’s A and B coefficients
are written as
B=
π e2 ω
πω
|1|r|2|2 =
|μ|2 ,
2
ε0 n r
ε0 n r 2
2nr ω 3
|μ|2 ,
A21 =
ε0 hc3
|μ|2 = |1|er|2|2 .
(2.39)
Note that many textbooks on quantum electronics use (2.39) as Einstein’s A
coefficient. Expressions on Einstein’s B coefficient depend on the definitions of
the stimulated emission rate whether it is proportional to the photon density or to the
energy density. In the preceding explanations, Einstein’s B coefficient is defined to
be in proportion to the photon density. Therefore, the quantum mechanical transition
rate for the stimulated emission is equal to Einstein’s B coefficient.
In semiconductors, due to the energy band structures, transitions take place
between various energy states, as shown in Fig. 2.8. If we put E 21 = E and
E 2 − E c = E , the electron energy in the valence band E for the allowed transition
2.4 Optical Gains
37
Fig. 2.8 Transition with a
constant photon energy
Ec
Ev
is given by E = E − E. Therefore, by integrating (2.27) with respect to E , the
net stimulated emission rate r 0 (stim) in the semiconductors is obtained as
r 0 (stim) =
e2 h n ph (E)
2m 2 ε0 n r 2 E
∞
|1|p|2|2 ρc (E )ρv (E )[ f 2 (E ) − f 1 (E )] dE ,
0
(2.40)
where (2.35) was used. Similarly, the optical power gain coefficient g(E) is given by
g(E) =
e2 h
2m 2 ε0 n r cE
∞
|1|p|2|2 ρc (E )ρv (E )[ f 2 (E ) − f 1 (E )] dE . (2.41)
0
From (2.10) and (2.36), the spontaneous emission rate r21 (spon) is expressed as
r21 (spon) =
4 π e2 n r E
m 2 ε0 h 2 c 3
∞
|1|p|2|2 ρc (E )ρv (E ) f 2 (E )[1 − f 1 (E )] dE .
0
(2.42)
When the excitation is weak, the absorption is observed, and the net absorption
rate r 0 (abs) is written as
r 0 (abs) = r12 (abs) − r21 (stim) = −r 0 (stim).
(2.43)
Therefore, the optical power absorption coefficient α(E) and the optical power gain
coefficient g(E) are related as
α(E) = −g(E).
(2.44)
Figure 2.9 shows the calculated optical power gain coefficient g(E) in (2.41) as a
function of E − E g for the bulk structures with the bandgap energy of E g = 0.8 eV.
With an increase in the carrier concentration n, the gain peak shifts toward a higher
energy (shorter wavelength). This shift of the gain peak is caused by the band filling
effect [2], in which the electrons in the conduction band and the holes in the valence
band occupy each band from the band edges. Because higher energy states are more
dense than lower energy states, the optical gain in higher energy states is larger than
that in lower energy ones.
38
2 Optical Transitions
Fig. 2.9 Optical power gain
coefficient
Optical Gain (cm−1)
600
400
n = 2.5 1018 cm−3
Eg = 0.8 eV
2.0
200
1.5
1.0
0
− 200
− 50
0
50
100
E − Eg (meV)
150
Here, let us consider the optical power gain coefficient g(E) from other viewpoints. In (2.41), ρc (E )ρv (E ) is considered to be the density of state pairs related
to the optical transitions. The density of state pairs is also expressed using the law
of momentum conservation (k-selection rule) and the law of energy conservation
(E 21 = E 2 − E 1 ). Under the k-selection rule, optical transitions take place only
for k = k1 = k2 . Therefore, we can define the density of state pairs or the reduced
density of states ρred (E 21 ) as
ρred (E 21 ) dE 21 ≡ ρv (E 1 ) dE 1 = ρc (E 2 ) dE 2 ,
(2.45)
dE 21 = dE 1 + dE 2 .
(2.46)
where
From (2.45) and (2.46), we have
1
1
+
ρred (E 21 ) =
ρv (E 1 ) ρc (E 2 )
−1
.
(2.47)
If the conduction and valence bands are parabolic, the energies E 1 and E 2 are
written as
2 k 2
,
2m h ∗
2 k 2
,
E2 = Ec +
2m e ∗
E1 = Ev −
(2.48)
(2.49)
where m h ∗ and m e ∗ are the effective masses of the holes and the electrons, respectively. As a result, E 1 and E 2 are rewritten as
2.4 Optical Gains
39
Fig. 2.10 Optical power
gain coefficient under the
k-selection rule
Optical Gain (cm−1)
600
400
n = 2.5 1018 cm−3
Eg = 0.8 eV
2.0
200
1.5
0
1.0
− 200
− 50
0
50
100
150
E − Eg (meV)
me∗
E 21 − E g ,
me∗ + mh∗
mh∗
E2 = Ec +
E 21 − E g .
∗
∗
me + mh
E1 = Ev −
(2.50)
(2.51)
Also, once the initial state of the optical transitions is given, the final state is
determined by E 21 and k. Hence, the number of state pairs per volume associated
with the optical transitions for the photon energies between E 21 and E 21 + dE 21 is
given by
(2.52)
ρred (E 21 ) [ f 2 (E 21 ) − f 1 (E 21 )] dE 21 .
We assume that the energy states related to optical transitions have an energy
width of /τin , and the spectral shape of Lorentzian such as
L(E 21 ) =
/τin
1
,
π (E 21 − E)2 + (/τin )2
(2.53)
where τin is the relaxation time due to electron scatterings and transitions. In this
case, the optical power gain coefficient g(E) is expressed as
e2 h
g(E) =
2m 2 ε0 n r cE
0
∞
|1|p|2|2 ρred (E 21 ) [ f 2 (E 21 ) − f 1 (E 21 )] L(E 21 ) dE 21 ,
(2.54)
which is plotted in Fig. 2.10 for the bulk structures with τin = 10−13 s. It is found
that relaxation is equivalent to band tailing, which contributes to optical transitions
in E < E g .
Because f 2 and [1 − f 1 ] are always positive, spontaneously emitted lights distribute in a higher energy than the gain peak. Figure 2.11 schematically shows spectra for
the stimulated emission (laser light) and the spontaneously emitted lights; its horizontal line indicates photon energy. Note that the photon energies increase due to the
band filling effect with enhancement of the excitation, irrespective of spontaneous
emission or stimulated emission.
Fig. 2.11 Stimulated emission and spontaneous emission
2 Optical Transitions
Light Intensity
40
Laser light
Spontaneous
emission
Photon Energy
References
1. Einstein, A.: Zur Quantentheorie der Strahlung. Phys. Z. 18, 121 (1917)
2. Pankove, J.I.: Optical Processes in Semiconductors. Dover, New York (1975)
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